Step 1: Find the equation of the line passing through $(\cos \theta, \sin \theta)$ and $(\cos \phi, \sin \phi)$ using two-point form: $(y - \sin \theta)(\cos \phi - \cos \theta) = (x - \cos \theta)(\sin \phi - \sin \theta)$

Step 2: Simplify to standard form $Ax + By + C = 0$: $x(\sin \phi - \sin \theta) + y(\cos \theta - \cos \phi) + \sin(\theta - \phi) = 0$ where $A = \sin \phi - \sin \theta$, $B = \cos \theta - \cos \phi$, and $C = \sin(\theta - \phi)$.

Step 3: Apply the perpendicular distance formula from origin $(0, 0)$: $d = \frac{|C|}{\sqrt{A^2 + B^2}} = \frac{|\sin(\theta - \phi)|}{\sqrt{(\sin \phi - \sin \theta)^2 + (\cos \theta - \cos \phi)^2}}$

Step 4: Simplify the denominator using trigonometric identities: $(\sin \phi - \sin \theta)^2 + (\cos \theta - \cos \phi)^2 = 2 - 2\cos(\phi - \theta) = 4\sin^2\left(\frac{\phi - \theta}{2}\right)$

Step 5: Therefore: $d = \frac{|\sin(\theta - \phi)|}{2\left|\sin\left(\frac{\phi - \theta}{2}\right)\right|} = \frac{\left|2\sin\left(\frac{\theta - \phi}{2}\right)\cos\left(\frac{\theta - \phi}{2}\right)\right|}{2\left|\sin\left(\frac{\theta - \phi}{2}\right)\right|} = \left|\cos\left(\frac{\theta - \phi}{2}\right)\right|$